Paths on a surface 
Sphere - fundamental group
In the sphere, as in the plane, any two paths with the same end points are homotopic. One way to think about this conclusion follows from the stereographic projection between the sphere and the plane: we fix a point in the sphere, reffered to as projection pole, and consider the tangent plane to the diametrically opposite point; to each point of the sphere we consider the line it defines together with the pole, and make this point correspond to the point of intersection between the line and the plane. This projection defines a one-to-one correspondence between the plane and the sphere (without the projection pole). Then, choosing a pole in the complement of the paths, and observing that the projection preserves paths, we can establish a correspondence between paths in the sphere and paths in the plane. Given two paths in the sphere, we can project them to the plane, consider the homotopy there existent between them, and project this homotopy back into the sphere between the two original paths.
Therefore, in the sphere, as in the plane, all paths are homotopic, and there is only one homotopy class.
Furthermore, for the case of the fundamental group, we can have that any closed path is homotopic to a constant path - which, can be taken as representative of the only homotopy class.
